The simple definition for centroid of a triangle is, the point of concurrency of the medians.

In the above triangle , AD, BE and CF are called medians. All the three medians AD, BE and CF are intersecting at G. So G is called centroid of the triangle

Example:1

Find the centroid of the triangle whose vertices are the points (8,4), (1,3) and (3,-1)

Solution:

The centroid of the triangle can be found by using the above formula.

Here, we have

(x

(x

(x

Plug the above values in to the formula

Centroid of the triangle = [(8+1+3)/3 , (4+3-1)/3]

= [12/3 , 6/3)

= (4 , 2)

Example:2

If a triangle has its centroid at (4,3) and two of its vertices are (2,-1) and (7,8). Find the third vertex.

Solution:

Let the third vertex be (a,b)

Here, we have

(x

(x

(x

Plug the above values in to the formula

Centroid of the triangle = (4,3)

[(2+7+a)/3 , (-1+8+b)/3] = (4 ,3)

[(9+a)/3 , (7+b)/3] = (4 , 3)

Equating the coordinates of x and y , we are getting

(9+a)/3 = 4 , (7+b)/3 = 3

9+a = 4(3) , 7+b = 3(3)

9+a = 12 , 7+b = 9

a = 12- 9 , b = 9-7

a = 3 , b= 2

Hence the third vertex (a , b) = (3 , 2)

This is how we are using the above formula to find centroid of a triangle and If centroid of the triangle and two of the vertices are given , we can find the third vertex.